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  "body": "\n## Example\n\nLet $p, q \\ge 0$ be integers and let $\\tau \\in \\C$ with $\\operatorname{Im} \\tau \\ne 0$.\nLet $W = (\\C^{p+1} \\setminus \\set{0}) \\times (\\C^{q+1} \\setminus \\set{0})$, and let the additive group $\\C$ act on $W$ by\n$$t \\cdot (u, v) = (e^{t} u, \\; e^{\\tau t} v).$$\nThe quotient $X_{p,q} = W / \\C$ is a *Calabi–Eckmann manifold*, a compact [[complex manifold|JSGWAN2RJV]] of dimension $p + q + 1$.\n\nCheck that $X_{p,q}$ is diffeomorphic to $S^{2p+1} \\times S^{2q+1}$: each orbit meets $S^{2p+1} \\times S^{2q+1}$ in exactly one point.\nWrite $t = x + iy$.\nThe conditions $\\abs{e^t u} = 1$ and $\\abs{e^{\\tau t} v} = 1$ read $x = -\\log\\abs{u}$ and $x \\operatorname{Re}\\tau - y \\operatorname{Im}\\tau = -\\log\\abs{v}$.\nSince $\\operatorname{Im}\\tau \\ne 0$, they have exactly one solution $(x, y)$.\nThe action is free and proper and is by holomorphic maps, so the quotient is a complex manifold of dimension $(p + 1) + (q + 1) - 1$.\n\n## Remarks\n\nThe map $[u, v] \\mapsto ([u], [v])$ makes $X_{p,q}$ a holomorphic [[fibre bundle|CX4M8K61AB]] over $\\mathbb{CP}^p \\times \\mathbb{CP}^q$ (see [[complex projective space|7GJ4STCNNM]]) whose fibres are [[elliptic curves|427N6S6QQW]].\nFor $p = q = 0$ one gets an elliptic curve, and for $p = 0$ a [[Hopf manifold|Z8R2D33WSM]] diffeomorphic to $S^1 \\times S^{2q+1}$.\nFor $p, q \\ge 1$ the manifold $S^{2p+1} \\times S^{2q+1}$ is simply connected and has $H^2 = 0$, so $X_{p,q}$ is not [[Kähler|X60JG9AEQJ]]: a compact Kähler manifold has a nonzero class in $H^2$.\nSo Calabi and Eckmann's construction gives simply connected compact complex manifolds that are not Kähler, for example on $S^3 \\times S^3$.\n\n## Explanations\n\n### L1 human\n\nA product of two odd-dimensional spheres, such as $S^3 \\times S^3$, has no obvious complex structure, since neither factor is even-dimensional.\nCalabi and Eckmann found one by viewing each sphere as a circle bundle over a complex projective space and combining the two circles into a complex torus.\nThe result is a compact complex manifold that cannot carry a Kähler metric.\n",
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